On Thursday, Trump tweeted a widely circulated parody video of Biden's apology for the incidents, which have threatened his early bid for the 2020 presidency.
"I see that you are on the job and presidential, as always," Biden tweeted in response.
Earlier in the day, the 76-year-old released a video promising to "be more mindful" of personal space. In the clip, Biden says, "I shake hands, I hug people, I grab men and women by the shoulders and say, 'You can do this.'"
Trump's parody video showed Biden sneaking up behind himself and touching himself even as he apologized. He captioned the tweet, "WELCOME BACK, JOE!"
Biden is expected to make an announcement about his plans for a presidential candidacy bid this month.
Biden currently faces allegations from multiple women that his physical behavior made them uncomfortable. Former Nevada state Assemblywoman Lucy Flores, the first woman to come forward, alleged that Biden kissed the back of her head at a 2014 Nevada campaign rally, making her feel "uneasy, gross and confused."
In his full video addressing his interaction with voters, released after statements from him and a campaign official, he emphasized that he wants to connect with people but also understands changing social norms.
"I've never thought of politics as cold and antiseptic," he said in Wednesday's video. "I've always thought it about connecting with people, as I said, shaking hands, hands on the shoulder, a hug, encouragement, and now, it's all about taking selfies together."
"You know, social norms have begun to change, they've shifted, and the boundaries protecting personal space have been reset, and I get it, I get it," he said.
While some Twitter users laughed at Trump's parody video, others called it out for being hypocritical, given the fact that the president has been accused by at least 15 women of misbehavior ranging from sexual harassment and sexual assault to lewd behavior around women. Most came forward in the wake of a 2005 Access Hollywood tape that was released in October 2016, in which he was caught saying on a hot microphone: "And when you're a star, they let you do it. You can do anything. ... Grab them by the p—. You can do anything."